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Then, by the well-ordering principle, there is a least element ; cannot be prime since a prime number itself is considered a length-one product of primes. By the definition of non-prime numbers, n {\displaystyle n} has factors a , b {\displaystyle a,b} , where a , b {\displaystyle a,b} are integers greater than one and less than n ...
Because 0 is the least element of N, it must be that 0 ∉ X. For any n ∈ N, suppose for every k ≤ n, k ∉ X. Then S(n) ∉ X, for otherwise it would be the least element of X. Thus, by the strong induction principle, for every n ∈ N, n ∉ X. Thus, X ∩ N = ∅, which contradicts X being a nonempty subset of N. Thus X has a least element.
The induction, bounding and least number principles are commonly used in reverse mathematics and second-order arithmetic. For example, I Σ 1 {\displaystyle {\mathsf {I}}\Sigma _{1}} is part of the definition of the subsystem R C A 0 {\displaystyle {\mathsf {RCA}}_{0}} of second-order arithmetic.
The least-upper-bound property is one form of the completeness axiom for the real numbers, and is sometimes referred to as Dedekind completeness. [2] It can be used to prove many of the fundamental results of real analysis , such as the intermediate value theorem , the Bolzano–Weierstrass theorem , the extreme value theorem , and the Heine ...
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...
In this setting, an ordered field K is Archimedean precisely when the following statement, called the axiom of Archimedes, holds: "Let x {\displaystyle x} be any element of K {\displaystyle K} . Then there exists a natural number n {\displaystyle n} such that n > x {\displaystyle n>x} ."
Similarly, the set of integers has the least-upper-bound property; if is a nonempty subset of and there is some number such that every element of is less than or equal to , then there is a least upper bound for , an integer that is an upper bound for and is less than or equal to every other upper bound for .
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα ( axíōma ), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.